Question:
If f(x) = sin (log x) and $y = f\left(\frac{2x+3}{3-2x}\right)$, then $\frac{dy}{dx}$ equals
If f(x) = sin (log x) and $y = f\left(\frac{2x+3}{3-2x}\right)$, then $\frac{dy}{dx}$ equals
Updated On: Jul 28, 2022
- $sin\left[log\left(\frac{2x+3}{3-2x}\right)\right]$
- $\frac{12}{\left(3-2x\right)^2}$
- $\frac{12}{\left(3-2x\right)^{2}}sin\left[log\left(\frac{2x+3}{3-2x}\right)\right]$
- $\frac{12}{\left(3-2x\right)^{2}}cos\left[log\left(\frac{2x+3}{3-2x}\right)\right]$
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The Correct Option is C
Solution and Explanation
Let $f'\left(x\right) = sin \left[log x\right]$ and $y = f\left(\frac{2x+3}{3-2x}\right)$
Now, $\frac{dy}{dx} = f' \left(\frac{2x+3}{3-2x}\right) . \frac{d}{dx}\left(\frac{2x+3}{3-2x}\right)$
$= sin\left[log\left(\frac{2x+3}{3-2x}\right)\right] \frac{\left[\left(6-4x-\right)-4x\left(-6\right)\right]}{\left(3-2x^{2}\right)}$
$= \frac{12}{\left(3-2x^{2}\right)}sin\left[log\left(\frac{2x+3}{3-2x}\right)\right]$
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Concepts Used:
Continuity & Differentiability
Definition of Differentiability
f(x) is said to be differentiable at the point x = a, if the derivative f ‘(a) be at every point in its domain. It is given by
Definition of Continuity
Mathematically, a function is said to be continuous at a point x = a, if
It is implicit that if the left-hand limit (L.H.L), right-hand limit (R.H.L), and the value of the function at x=a exist and these parameters are equal to each other, then the function f is said to be continuous at x=a.
If the function is unspecified or does not exist, then we say that the function is discontinuous.